Digital SAT · Solve it in Desmos

How to Solve Linear Inequalities on Desmos (Digital SAT)

Q: Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Q: Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

The shaded region confirms the direction of the sign, the thing students most often flip. Solve by hand on simple ones and use the graph as a check when the negative divide worries you.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for linear inequalities

The Desmos steps

1
Type the inequality
Enter it with x to see the shaded solution range.
2
Find the edge
The boundary is the edge of the shaded region; note whether it is included.
3
Pick the value
Read the greatest or least value, or the greatest integer, from the shaded range.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example

5x - 4 \leq 26

, what is the greatest possible value of

x

5

6

22

30

A: Incorrect. This ignores the $-4$ and uses $26 \div 5$ rounded down to 5.

B: Correct. $5x \leq 30$ gives $x \leq 6$ , so the greatest value is 6.

C: Incorrect. This forgets to divide by 5: $26 - 4 = 22$ .

D: Incorrect. This forgets to divide by 5 after adding: $26 + 4 = 30$ .

Explanation

Add 4 to both sides to get $5x \leq 30$ , then divide by 5: $x \leq 6$ ; the greatest value is 6.

Medium example

What is the greatest integer value of

x

that satisfies

4x + 9 < 30

4

5

6

21

A: Incorrect. This subtracts an extra 1 unnecessarily from the correct bound.

B: Correct. $4x < 21$ gives $x < 5.25$ , so the greatest integer is 5.

C: Incorrect. This rounds 5.25 up instead of down for a strict inequality.

D: Incorrect. This forgets to divide by 4: $30 - 9 = 21$ .

Explanation

Subtract 9: $4x < 21$ . Divide by 4: $x < 5.25$ . The greatest integer less than 5.25 is 5.

Hard example

y > x + 2

and

y < 4x - 1

Which of the following tables gives three values of x and their corresponding values of y that are all solutions to the given system of inequalities?

x	y
2	0
3	1
4	2

x	y
2	6
3	8
4	11

x	y
2	6
3	8
4	1

x	y
2	7
3	11
4	15

A: The pair $(x, y) = (2, 0)$ does not satisfy both inequalities in the system, so not all values in this table are solutions.

B: Every ordered pair in this table satisfies both inequalities in the system, so all three values of x and y are solutions.

C: The pair $(x, y) = (4, 1)$ does not satisfy both inequalities in the system, so not all values in this table are solutions.

D: The pair $(x, y) = (2, 7)$ does not satisfy both inequalities in the system, so not all values in this table are solutions.

Explanation

Substitute each ordered pair (x, y) from a table into both inequalities in the system. A table is correct only when all three of its pairs are solutions. The correct table is the one whose every pair makes both inequalities in the system true; any table containing even one pair that fails is incorrect.

Try it with the calculator

Question 1easy

A delivery driver can carry at most 12 packages at one time and is already carrying 7 packages. What is the greatest number of additional packages the driver can carry at the same time?

Question 2easy

A food cart can serve at most 90 customers in one day. It serves $b$ customers in the morning and $l$ customers in the afternoon. Which inequality represents this situation?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve linear inequalities questions on the SAT?

The shaded region confirms the direction of the sign, the thing students most often flip. Solve by hand on simple ones and use the graph as a check when the negative divide worries you.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master linear inequalities, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full method Take the free diagnostic

The Desmos steps

Type the inequality

Find the edge

Pick the value

Worked in Desmos

Try it with the calculator

Question 3 is ready when you are

Desmos questions

Can Desmos solve linear inequalities questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Should I still learn the algebra?

Master linear inequalities, calculator and all